Vortex rings for the Gross-Pitaevskii equation

Abstract

We provide a mathematical proof of the existence of traveling vortex rings solutions to the Gross-Pitaevskii (GP) equation in dimension N ≥ 3. We also extend the asymptotic analysis of the free field Ginzburg-Landau equation to a larger class of equations, including the GinzburgLandau equation for superconductivity as well as the traveling wave equation for GP. In particular we rigorously derive a curvature equation for the concentration set (i.e. line vortices if N = 3).